Global Tangentially Analytical Solutions of the 3D Axially Symmetric Prandtl Equations

Abstract

In this paper, the authors will prove the global existence of solutions to the three dimensional axially symmetric Prandtl boundary layer equations with small initial data, which lies in H ¹ Sobolev space with respect to the normal variable and is analytical with respect to the tangential variables. The main novelty of this paper relies on careful constructions of a tangentially weighted analytic energy functional and a specially designed good unknown for the reformulated system. The result extends that of Paicu-Zhang in [Paicu, M. and Zhang, P., Global existence and the decay of solutions to the Prandtl system with small analytic data, Arch. Ration. Mech. Anal., 241(1), 2021, 403–446]. from the two dimensional case to the three dimensional axially symmetric case, but the method used here is a direct energy estimates rather than Fourier analysis techniques applied there.

Keywords

Global existence / Tangentially analytical solutions / Axially symmetric / Prandtl equations

References

[1] Albritton D, Brué E, Colombo M. Non-uniqueness of Leray solutions of the forced Navier-Stokes equations. Ann. of Math. (2), 2022, 196(1): 415-455

[2] Alexandre R, Wang Y G, Xu C J, Yang T. Well-posedness of the Prandtl equation in Sobolev spaces. J. Amer. Math. Soc., 2015, 28(3): 745-784

[3] Carrillo B, Pan X, Zhang Q S, Zhao Z. Decay and vanishing of some D-solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal., 2020, 237(3): 1383-1419

[4] Chen C C, Strain R M, Tsai T P, Yau H T. Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations II. Comm. Partial Differential Equations, 2009, 34(1–3): 203-232

[5] Dietert, H. and Gérard-Varet, D., Well-posedness of the Prandtl equations without any structural assumption, Ann. PDE, 5(1), 2019, 51 pp.

[6] E W, Engquist B. Blowup of solutions of the unsteady Prandtl’s equation. Comm. Pure Appl. Math., 1997, 50(12): 1287-1293

[7] Gérard-Varet D, Dormy E. On the ill-posedness of the Prandtl equation. J. Amer. Math. Soc., 2010, 23(2): 591-609

[8] Gérard-Varet D, Masmoudi N. Well-posedness for the Prandtl system without analyticity or monotonicity. Ann. Sci. Éc. Norm. Supér., 2015, 48(4): 1273-1325

[9] Gérard-Varet D, Nguyen T. Remarks on the ill-posedness of the Prandtl equation. Asymptot. Anal., 2012, 77(1–2): 71-88

[10] Guo Y, Nguyen T. A note on Prandtl boundary layers. Comm. Pure Appl. Math., 2011, 64(10): 1416-1438

[11] Hörmander L. The analysis of linear partial differential operators III, Pseudodifferential operators, 1985, Berlin: Springer-Verlag

[12] Ignatova M, Vicol V. Almost global existence for the Prandtl boundary layer equations. Arch. Ration. Mech. Anal., 2016, 220(2): 809-848

[13] Koch G, Nadirashvili N, Seregin G A, Šverák V. Liouville theorems for the Navier-Stokes equations and applications. Acta Math., 2009, 203(1): 83-105

[14] Kukavica I, Vicol V. On the local existence of analytic solutions to the Prandtl boundary layer equations. Commun. Math. Sci., 2013, 11(1): 269-292

[15] Li W X, Yang T. Well-posedness in Gevrey function spaces for the Prandtl equations with non-degenerate critical points. J. Eur. Math. Soc. (JEMS), 2020, 22(3): 717-775

[16] Li W X, Masmoudi N, Yang T. Well-posedness in Gevrey function space for 3D Prandtl equations without Structural Assumption. Comm. Pure Appl. Math., 2022, 75(8): 1755-1797

[17] Lin X, Zhang T. Almost global existence for the 3D Prandtl boundary layer equations. Acta Appl. Math., 2020, 169: 383-410

[18] Liu J G, Wang W C. Characterization and regularity for axisymmetric solenoidal vector fields with application to Navier-Stokes equation. SIAM J. Math. Anal., 2009, 41(5): 1825-1850

[19] Liu C J, Wang Y G, Yang T. On the ill-posedness of the Prandtl equations in three-dimensional space. Arch. Ration. Mech. Anal., 2016, 220(1): 83-108

[20] Liu C J, Wang Y G, Yang T. A well-posedness theory for the Prandtl equations in three space variables. Adv. Math., 2017, 308: 1074-1126

[21] Lombardo M C, Cannone M, Sammartino M. Well-posedness of the boundary layer equations. SIAM J. Math. Anal., 2003, 35(4): 987-1004

[22] Masmoudi N, Wong T K. Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods. Comm. Pure Appl. Math., 2015, 68(10): 1683-1741

[23] Oleinik O A, Samokhin V N. Mathematical models in boundary layer theory, 1999, Boca Raton, FL: Chapman & Hall/CRC

[24] Paicu M, Zhang P. Global existence and the decay of solutions to the Prandtl system with small analytic data. Arch. Ration. Mech. Anal., 2021, 241(1): 403-446

[25] Pan X. Regularity of solutions to axisymmetric Navier-Stokes equations with a slightly supercritical condition. J. Differential Equations, 2016, 260(12): 8485-8529

[26] Prandtl L. Über Flüssigleitsbewegung bei sehr kleiner Reibung. Verhandlung des III Intern. Math. Kongresses, Heidelberg, 1904 484-491

[27] Sammartino M, Caflisch R E. Zero viscosity limit for analytic solutions, of the Navier-Stokes equation on a half-space I, Existence for Euler and Prandtl equations. Comm. Math. Phys., 1998, 192(2): 433-461

[28] Wang, C., Wang, Y. and Zhang, P., On the global small solution of 2-D Prandtl system with initial data in the optimal Gevrey class, arXiv: 2103.00681

[29] Xin Z, Zhang L. On the global existence of solutions to the Prandtl’s system. Adv. Math., 2004, 181(1): 88-133

[30] Xin, Z., Zhang, L. and Zhao, J., Global Well-posedness and Regularity of Weak Solutions to the Prandtl’s System, arXiv: 2203.08988.

[31] Xu C J, Zhang X. Long time well-posedness of Prandtl equations in Sobolev space. J. Differential Equations, 2017, 263(12): 8749-8803

[32] Zhang P, Zhang Z. Long time well-posedness of Prandtl system with small and analytic initial data. J. Funct. Anal., 2016, 270(7): 2591-2615